Quantum Zeno Effect in Heisenberg Picture and Critical Measurement Time

Quantum Zeno effect is conventionally interpreted by the assumption of the wave-packet collapse, in which does not involve the duration of measurement. However, we predict duration $\tau_m$ of each measurement will appear in quantum Zeno effect by a dynamical approach. Moreover, there exists a model-free critical measurement time, which quantum Zeno effect does not occur when $\tau_m$ takes some special values. In order to give these predictions, we first present a description of quantum Zeno effect in the Heisenberg picture, which is based on the expectation value of an observable and its fluctuation. Then we present a general proof for quantum Zeno effect in the Heisenberg picture, which is independent of the concrete systems. Finally, we calculate the average population and relative fluctuation after $N$ successive measurements in XX model, which agrees with our prediction about the critical measurement time.